A key property of neural networks is their capacity of adapting to data during training. Yet, our current mathematical understanding of feature learning and its relationship to generalization remain limited. In this work, we provide a random matrix analysis of how fully-connected two-layer neural networks adapt to the target function after a single, but aggressive, gradient descent step. We rigorously establish the equivalence between the updated features and an isotropic spiked random feature model, in the limit of large batch size. For the latter model, we derive a deterministic equivalent description of the feature empirical covariance matrix in terms of certain low-dimensional operators. This allows us to sharply characterize the impact of training in the asymptotic feature spectrum, and in particular, provides a theoretical grounding for how the tails of the feature spectrum modify with training. The deterministic equivalent further yields the exact asymptotic generalization error, shedding light on the mechanisms behind its improvement in the presence of feature learning. Our result goes beyond standard random matrix ensembles, and therefore we believe it is of independent technical interest. Different from previous work, our result holds in the challenging maximal learning rate regime, is fully rigorous and allows for finitely supported second layer initialization, which turns out to be crucial for studying the functional expressivity of the learned features. This provides a sharp description of the impact of feature learning in the generalization of two-layer neural networks, beyond the random features and lazy training regimes.

Understanding the advantages of deep neural networks trained by gradient descent (GD) compared to shallow models remains an open theoretical challenge. In this paper, we introduce a class of target functions (single and multi-index Gaussian hierarchical targets) that incorporate a hierarchy of latent subspace dimensionalities. This framework enables us to analytically study the learning dynamics and generalization performance of deep networks compared to shallow ones in the high-dimensional limit. Specifically, our main theorem shows that feature learning with GD successively reduces the effective dimensionality, transforming a high-dimensional problem into a sequence of lower-dimensional ones. This enables learning the target function with drastically less samples than with shallow networks. While the results are proven in a controlled training setting, we also discuss more common training procedures and argue that they learn through the same mechanisms. These findings open the way to further quantitative studies of the crucial role of depth in learning hierarchical structures with deep networks.

We study the impact of the batch size $n_b$ on the iteration time $T$ of training two-layer neural networks with one-pass stochastic gradient descent (SGD) on multi-index target functions of isotropic covariates. We characterize the optimal batch size minimizing the iteration time as a function of the hardness of the target, as characterized by the information exponents. We show that performing gradient updates with large batches $n_b \lesssim d^{\frac{\ell}{2}}$ minimizes the training time without changing the total sample complexity, where $\ell$ is the information exponent of the target to be learned and $d$ is the input dimension. However, larger batch sizes than $n_b \gg d^{\frac{\ell}{2}}$ are detrimental for improving the time complexity of SGD. We provably overcome this fundamental limitation via a different training protocol, Correlation loss SGD, which suppresses the auto-correlation terms in the loss function. We show that one can track the training progress by a system of low-dimensional ordinary differential equations (ODEs). Finally, we validate our theoretical results with numerical experiments.

In this manuscript, we investigate the problem of how two-layer neural networks learn features from data, and improve over the kernel regime, after being trained with a single gradient descent step. Leveraging the insight from (Ba et al., 2022), we model the trained network by a spiked Random Features (sRF) model. Further building on recent progress on Gaussian universality (Dandi et al., 2023), we provide an exact asymptotic description of the generalization error of the sRF in the high-dimensional limit where the number of samples, the width, and the input dimension grow at a proportional rate. The resulting characterization for sRFs also captures closely the learning curves of the original network model. This enables us to understand how adapting to the data is crucial for the network to efficiently learn non-linear functions in the direction of the gradient - where at initialization it can only express linear functions in this regime.

We investigate the training dynamics of two-layer neural networks when learning multi-index target functions. We focus on multi-pass gradient descent (GD) that reuses the batches multiple times and show that it significantly changes the conclusion about which functions are learnable compared to single-pass gradient descent. In particular, multi-pass GD with finite stepsize is found to overcome the limitations of gradient flow and single-pass GD given by the information exponent (Ben Arous et al., 2021) and leap exponent (Abbe et al., 2023) of the target function. We show that upon re-using batches, the network achieves in just two time steps an overlap with the target subspace even for functions not satisfying the staircase property (Abbe et al., 2021). We characterize the (broad) class of functions efficiently learned in finite time. The proof of our results is based on the analysis of the Dynamical Mean-Field Theory (DMFT). We further provide a closed-form description of the dynamical process of the low-dimensional projections of the weights, and numerical experiments illustrating the theory.

For high-dimensional Gaussian data, we investigate theoretically how the features of a two-layer neural network adapt to the structure of the target function through a few large batch gradient descent steps, leading to an improvement in the approximation capacity with respect to the initialization. First, we compare the influence of batch size to that of multiple (but finitely many) steps. For a single gradient step, a batch of size $n = O(d)$ is both necessary and sufficient to align with the target function, although only a single direction can be learned. In contrast, $n = O(d^2)$ is essential for neurons to specialize in multiple relevant directions of the target with a single gradient step. Even in this case, we show there might exist “hard” directions requiring $n = O(d^\ell)$ samples to be learned, where $\ell$ is known as the leap index of the target. Second, we show that the picture drastically improves over multiple gradient steps: a batch size of $n = O(d)$ is indeed sufficient to learn multiple target directions satisfying a staircase property, where more and more directions can be learned over time. Finally, we discuss how these directions allow for a drastic improvement in the approximation capacity and generalization error over the initialization, illustrating a separation of scale between the random features/lazy regime and the feature learning regime. Our technical analysis leverages a combination of techniques related to concentration, projection-based conditioning, and Gaussian equivalence, which we believe are of independent interest. By pinning down the conditions necessary for specialization and learning, our results highlight the intertwined role of the structure of the task to learn, the detail of the algorithm (the batch size), and the architecture (i.e., the number of hidden neurons), shedding new light on how neural networks adapt to the feature and learn complex task from data over time.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allows us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussions in the literature about Gaussian universality of the errors in this context.

A simple model to study subspace clustering is the high-dimensional $k$-Gaussian mixture model where the cluster means are sparse vectors. Here we provide an exact asymptotic characterization of the statistically optimal reconstruction error in this model in the high-dimensional regime with extensive sparsity, i.e. when the fraction of non-zero components of the cluster means $\rho$, as well as the ratio $\alpha$ between the number of samples and the dimension are fixed, while the dimension diverges. We identify the information-theoretic threshold below which obtaining a positive correlation with the true cluster means is statistically impossible. Additionally, we investigate the performance of the approximate message passing (AMP) algorithm analyzed via its state evolution, which is conjectured to be optimal among polynomial algorithm for this task. We identify in particular the existence of a statistical-to-computational gap between the algorithm that requires a signal-to-noise ratio $\lambda_{\text{alg}} \ge k / \sqrt{\alpha}$ to perform better than random, and the information theoretic threshold at $\lambda_{\text{it}} \approx \sqrt{-k \rho \log{\rho}} / \sqrt{\alpha}$. Finally, we discuss the case of sub-extensive sparsity $\rho$ by comparing the performance of the AMP with other sparsity-enhancing algorithms, such as sparse-PCA and diagonal thresholding.